2. a. Match each ratio from column I with the equivalent ratio in column II.
Exercise numbers appearing in color are answered in the Selected Answers appendix.
b. Write each matched pair of equivalent ratios as a percent.
3. Your biology instructor returned three quizzes today. On which quiz did you perform best?
Explain how you determined the best score.
Quiz 1: 18 out of 25 Quiz 2: 32 out of 40 Quiz 3: 14 out of 20
4. Baseball batting averages are the ratios of hits to ‘at bats.’ They are reported as three-digit decimals. Determine the batting averages and ranking of three players with the following records.
a. 16 hits out of 54 at bats b. 25 hits out of 80 at bats
c. 32 hits out of 98 at bats
5. There are 1720 women among the 3200 students at the local community college. Express this ratio in each of the following forms:
a. fraction b. reduced fraction
d. percent c. decimal
6. At the state university campus, there are 2304 women and 2196 men enrolled. In which school, the community college in Exercise 5 or the state university, is the relative number of women greater? Explain your reasoning.
7. In the 2008 Major League Baseball season, the world champion Philadelphia Phillies ended their regular season with a 92–70 win-loss record. During the playoff season, their win-loss record was 10–4. Did they play better in the regular season or in the postseason? Justify your answer mathematically.
8. A random check of 150 Southwest Airlines flights last month identified that 113 of them arrived on time. What “on-time” percent does this represent?
MATRICULATION BY GENDER FULL-TIME MATRICULATED
(»12 CREDITS)
PART-TIME MATRICULATED ( 12 CREDITS)
<
PART-TIME NONMATRICULATED (<12 CREDITS)
Men 214 174 65
Women 262 87 29
9. A consumer magazine reported that of the 13,350 subscribers who owned brand A dish-washer, 2940 required a service call. Only 730 of the 1860 subscribers who owned brand B needed dishwasher repairs. Which brand dishwasher has the better repair record?
10. In the following table, the admissions office has organized data describing the number of men and women who are currently enrolled in your college in order to formulate recruitment and retention efforts for the coming academic year. A student is
matriculated if he or she is enrolled in a program that leads to a degree (such as associ-ate or baccalaureassoci-ate).
a. How many men attend your college?
b. What percent of the men are full-time students?
c. How many women attend your college?
d. What percent of the women are full-time students?
e. How many students are enrolled full-time?
f. How many students are enrolled part-time?
g. Women constitute what percent of the full-time students?
h. Women constitute what percent of the part-time students?
i. How many students are nonmatriculated?
j. What percent of the student body is nonmatriculated?
Definition
Proportional reasoning is the thought process by which you apply a known ratio to one piece of information (part or total) to determine a related, but yet unknown, piece of in-formation (total or part).
The following describes the direct method of applying a known ratio to a given piece of information. known ratio to a given piece of information.
The data in the paragraph above is typical of the kind of numer-ical information you will find in reading virtually any printed document, report, or article.
1. What relative data (ratio) appears in the opening paragraph?
What phrase or symbol identifies it as a relative measure?
2. Express the ratio in Problem 1 in fractional, decimal, and verbal form.
Recall from Activity 1.6 that a ratio represents a relative measure, the quotient of a part divided by a total.
3. The opening paragraph contains a second piece of information, namely, 1,000,000. Does this number represent a part or a total in this situation?
Once you know two of the three values in the basic relationship
the third unknown value can be determined using proportional reasoning.
ratio = part total,
Procedure
Applying the Known Ratio Directly by Multiplication/Division Step 1: Identify the known ratio.
Step 2: Identify whether the given piece of information represents a total or a part.
Step 3: Rewrite the basic relationship, in one of two equivalent forms:
• If the given information represents a total, then the part is unknown. You can determine the unknown part by multiplying the known total by the known ratio as follows:
• If the given information represents a part, then the total is the unknown
In this case, you divide the known part by the known ratio to determine the total.
unknowntotal = part , ratio.
unknownpart = total # ratio.
ratio = part total,
International public health experts are desperately seeking to stem the spread of the AIDS epidemic in sub-Saharan Africa, espe-cially in the small nation of Botswana. According to a recent United Nations report, approximately 23.9% of Botswana’s 1,000,000 adults (ages 15–49) were infected with the AIDS virus.
Botswana
Examples 1 and 2 demonstrate this procedure.
4. In 2007, sub-Saharan Africa was home to approximately 22,000,000 people living with HIV, which is 67% of all people living with HIV. From this, estimate the number of peo-ple in the world living with HIV in 2007.
5. In 2008 the U.S. Census Bureau estimated the U.S. population at approximately 304 million. Approximately 27% of this population was under the age of 20 years. How many people were under the age of 20 in the United States in 2008?
Example 1 How many adults in Botswana are infected with AIDS (refer to the introductory paragraph)?
SOLUTION
Proportional reasoning is needed to determine the actual number of adults in Botswana who are infected with AIDS.
Step 1: The known ratio is 23.9%, or 0.239 in decimal form—that is, 239 out of every 1000 adults.
Step 2: The given piece of information is Botswana’s total adult population, 1,000,000.
Step 3: The unknown piece of information is the actual number, or part, of Botswana’s adults infected with AIDS. Therefore,
The number of adults in Botswana infected with AIDS is 239,000.
unknownpart = 1,000,000# 0.239 = 239,000.
unknownpart = total # ratio,
Example 2 You do further research about this disturbing health problem and discover that, in 2007, there were approximately 22,000,000 adults and children living with HIV/AIDS in sub-Saharan Africa. This rep-resented 5% of the total population there. Use proportional reasoning to determine the total population of sub-Saharan Africa in 2007.
SOLUTION
The known ratio is 5% or 0.05.
The given information, 22,000,000 adults and children with HIV/AIDS, is a part of the total population.
.
The total population of sub-Saharan Africa is approximately 440 million people.
unknowntotal = 22,000,000
0.05 = 440,000,000 unknowntotal = part , ratio,